Alas, there is no way to change the faction type once it's created. I think we may consider removing public faction option altogether as it's rarely used and causes confusion to some players.
You can create a new private faction and invite your most active members there. If there are any faction-owned land claims, they must be deconstructed and rebuilt (all claims of the whole base at once) as it's not possible to transfer them from one faction to another.
Enjoy the game!